Shortest Watchman Tours in Simple Polygons under Rotated Monotone Visibility

TL;DR

This paper introduces an efficient algorithm to compute the shortest watchman tour in simple polygons considering rotated monotone visibility, optimizing for the minimal tour length as the viewing direction varies.

Contribution

It presents an $O(nrG)$ time algorithm for dynamic computation of minimal watchman tours under rotated monotone visibility in simple polygons.

Findings

Algorithm runs in $O(nrG)$ time.

Effectively maintains minimal watchman tours as direction varies.

Provides a method to optimize polygon surveillance paths.

Abstract

We present an $O(nrG)$ time algorithm for computing and maintaining a minimum length shortest watchman tour that sees a simple polygon under monotone visibility in direction $\theta$, while $\theta$ varies in $[0,180^{\circ})$, obtaining the directions for the tour to be the shortest one over all tours, where $n$ is the number of vertices, $r$ is the number of reflex vertices, and $G\leq r$ is the maximum number of gates of the polygon used at any time in the algorithm.